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For a stationary circuit, E has been given in Article 69 as C E dl, where E is the static “electric 4 intensity, or electromotive force”. For a moving circuit, E can be written in terms of aR vector Ev as [1, art. 598, eq. (6)], [2, sec. 6.1]

E(t)= Ev(r,t) · dl (4) I C(t) where C(t) denotes the closed curve of the moving circuit and Ev(r,t) is the unknown unit-charge force that would be exerted on a hypothetical5 electric charge placed at and moving with the point r of the circuit (see Articles 68 and 598). The vector form of the line integral in (4) is given in [1, art. 598, eq. (6)]. (It should be noted that in Article 579, E represents the − − · “impressed” voltage produced by a battery in the circuit so that E IR in Article 579 equals C(t) Ev dl and thus in Article 579 Maxwell writes E − IR = dp/dt.) H The p(t) in (3) is given in Articles 590 and 591 in terms of the vector potential A(r,t) and magnetic induction field B(r,t)

p(t)= A(r,t) · dl = B(r,t) · ˆn dS (5) I Z C(t) S(t) with S(t) any open surface bounded by the circuit (closed curve) C(t) and l (Maxwell’s ρ) is “the vector from the origin to a point of the circuit.” The vector ˆn is the usual surface unit normal defined by the right-hand rule with respect to the direction dl around the curve C(t). On the curve C(t), the vectors r and l are identical (r = l on C(t)). The vector forms of the line and surface integrals in (5) are given in [1, art. 590, eq. (7)] and [1, art. 591, eq. (12)], respectively. With (4) and (5) inserted into (3), we see that Maxwell has obtained the most general integral form of Faraday’s law d d Ev(r,t) · dl = − A(r,t) · dl = − B(r,t) · ˆn dS. (6) I dt I dt Z C(t) C(t) S(t)

When he writes (3) in Article 595, however, he has not yet shown for moving circuits that the electromotive force (Ev) on a hypothetical unit charge moving with the circuit at r is equal to what now is generally called the Lorentz force E + v × B per moving unit electric charge, where v is the velocity of the moving charge. (Before Maxwell evaluates Ev to show that it equals E + v × B, he has inferred that Ev is the electromotive force on a moving unit electric charge from Faraday’s measurements [6] of voltages induced in circuits moving through magnetic fields.) Maxwell allowed his moving “circuit” C(t) in (6) to be a purely geometrical closed curve moving through the free-space vacuum/ether or probably even through (not just with) a conductor or polarized material. For example, in Article 598 he · references Article 69, where he defines C E dl for the static electric field and explains that this integral is “the work that would be done by the electric force onR a unit of positive electricity [electric charge] carried along the curve from A, the beginning, to P , the end of the arc.” Moreover, in Article 586, Maxwell says, “we are not now considering a current the parts of which may, and indeed do, act on one another, but a mere circuit, that is, a closed curve along which a current may flow, and this is a purely geometrical figure, the parts of which cannot be conceived to have any physical action on each other.” Similar use of the term “circuit” to refer to any geometrical “closed curve” is found in Articles 587–591, which lead into Article 598. For stationary circuits, he confirms toward the end of Article 598 that, as in Article 69

E(t)= E(r,t) · dl (7) I C where E(r,t) is the electric force on a hypothetical unit electric charge placed at r as explained in Article 68. Consequently, Maxwell has obtained the integral form of Faraday’s law for stationary circuits, namely ∂ ∂ E(r,t) · dl = − A(r,t) · dl = − B(r,t) · ˆn dS. (8) I I ∂t Z ∂t C C S

4Maxwell uses the same symbol E for the force on a stationary or moving unit electric charge. Here, as in [2], the symbol E is reserved for the force on a stationary unit electric charge (the usual electric field) and the symbol Ev is used for the force on a unit electric charge moving with velocity v. (Of course, Ev=0 = E.) Also, it should be noted that J. J. Thomson evidently changed the term “electromotive force” used by Maxwell in Article 618 and elsewhere in volume two of the first and second editions of Maxwell’s Treatise to “electromotive intensity” in the third edition of Maxwell’s Treatise. Whenever this occurs in the quotes of Maxwell used in this paper, I have reinserted Maxwell’s original word “force”. 5The adjective “hypothetical” is used herein to denote the force that would be exerted on a small particle carrying a unit of electric charge if it were placed at a point without disturbing the given sources. In the words that Maxwell used to define the static electric field with a stationary unit charge, “The resultant electric intensity [field] at any point is the force which would be exerted on a small body charged with the unit of positive electricity, if it were placed there without disturbing the actual distribution of electricity” [1, art. 68]. Application of Stokes’ theorem to (8) yields the differential form of Faraday’s law ∂ ∂ ∇× E(r,t)= −∇× A(r,t)= − B(r,t). (9) ∂t ∂t However, Maxwell does not write this differential form of Faraday’s law in his Treatise nor in his 1865 paper [9] which contain only the integral form of Faraday’s law.6 The first equation in (8) or (9) implies that ∂ E(r,t)= − A(r,t) − ∇ψ (r,t) (10) ∂t e for stationary circuits, where ψe(r,t) is a time-dependent as well as a spatially dependent scalar potential function. In Article 598 Maxwell says that ψe(r,t) “represents, according to a certain definition, the electric potential,” which he later says in 2 Article 783 satisfies Poisson’s equation ∇ ψe(r,t) = −ρ(r,t)/ǫ in a homogeneous isotropic dielectric with permittivity ǫ. This Poisson equation follows from Maxwell using the Coulomb gauge ∇ · A = 0 throughout his Treatise [2, secs. 6.2 and 6.4]. Maxwell also writes (10) explicitly as his second equation in Article 783 for time varying fields with v =0.

III. DERIVATION OF THE FORCE ON A MOVING UNIT ELECTRIC CHARGE Returning to Maxwell’s general integral form of Faraday’s law for moving circuits in (3), expressed more fully in (6), we − · − − find in Article 598 Maxwell’s ingenious evaluation of (d/dt) C(t) A(r,t) dl to prove that Ev(r,t) = ∂A(r,t)/∂t ∇ψe(r,t)+ v(r,t) × B(r,t), where v(r,t) is the velocity at eachH point r = l of the moving circuit C(t). He thus completes the mathematical formulation of Faraday’s integral law for moving circuits and in so doing derives the force exerted on a moving unit electric charge by the magnetic induction field B. Maxwell accomplishes this feat as follows. He writes A(r,t) · dl in rectangular coordinates as ∂x ∂y ∂z A(r,t) · dl = A ds + A ds + A ds (11) x ∂s y ∂s z ∂s where s is defined by Maxwell in Articles 16 and 69 as “the length of the arc, measured from A [the initial point on the arc]”. Thus, ds is the scalar element of length on the closed curve C(t) at a fixed time t. The x, y, and z are the rectangular components of the position vector l = r = xˆx + yˆy + zˆz on C(t). These rectangular components are functions of the time t and the length s along the curve C(t), that is l(t,s)= r(t,s)= x(t,s)ˆx + y(t,s)ˆy + z(t,s)ˆz, l = r ∈ C(t). (12) If we consider the integral ∂x ∂y ∂z A(r,t) · dl = Ax + Ay + Az ds (13) I I  ∂s ∂s ∂s  C(t) C(t) ′ one can change the scalar integration variable s to s = s/smax at each instant of time t, where smax is the total length of the closed curve C(t) at the time t, and (13) becomes 1 ∂x ∂y ∂z ′ A(r,t) · dl = Ax + Ay + Az ds (14) I I  ∂s′ ∂s′ ∂s′  C(t) 0 and thus the limits of the integration variable s′ need not change with time t. Since C(t) is a closed curve, s′ =0 and s′ =1 refer to the same point on C(t). Maxwell didn’t do this renormalization of the s variable explicitly because it was probably obvious to him that the limits of the integration variable s can be chosen to be independent of the time variable t since it occurs in both the numerator and denominator of the right hand side of (11).

6We know that Maxwell deliberately chose to emphasize the integral form of Faraday’s law in his Treatise and 1865 paper [9] since he had deduced the differential form of this law from his “theory of molecular vortices” that he used in his 1861 paper [8] to explain Faraday’s experimental results [10]. In Part 2 of that 1861 paper, which contains no integrals, Maxwell wrote the scalar version of ∇× E = −µ∂H/∂t (= −∂B/∂t) as his equation (54). Maxwell’s mathematical formulation of Faraday’s law is not contained in Maxwell’s earlier 1856 paper [11] in either the integral or differential form. Taking the time derivative of this equation, we can bring the time derivative of the right-hand side under the integral sign (because the limits of integration do not depend on time) to get 1 d d ∂x A(r,t) · dl = Ax r(t,s),t + ··· ds dt I I dt  ∂s  C(t) 0   1 ∂A (r,t) ∂x ∂A (r,t) ∂y ∂A (r,t) ∂z = x + y + z I  ∂t ∂s ∂t ∂s ∂t ∂s 0 ∂A ∂x ∂A ∂y ∂A ∂z ∂x + x + y + z (15)  ∂x ∂s ∂x ∂s ∂x ∂s  ∂t ∂A ∂x ∂A ∂y ∂A ∂z ∂y + x + y + z  ∂y ∂s ∂y ∂s ∂y ∂s  ∂t ∂A ∂x ∂A ∂y ∂A ∂z ∂z + x + y + z  ∂z ∂s ∂z ∂s ∂z ∂s  ∂t ∂2x ∂2y ∂2z + Ax(r,t) + Ay(r,t) + Az(r,t) ds  ∂s∂t ∂s∂t ∂s∂t where the superfluous prime on the integration variable s′ has been dropped. The partial derivatives with respect to s are taken holding t fixed. The partial derivatives of (x,y,z) with respect to t are taken holding s fixed. The partial derivatives of A(r,t) with respect to x, y, or z are taken holding t fixed and the partial t derivative of A(r,t) is taken holding (x,y,z) fixed. To obtain (15), use has been made of the chain rules dA(r,t) ∂A(r,t) ∂A(r,t) dx ∂A dy ∂A dz = + + + (16) dt ∂t ∂x dt ∂y dt ∂z dt dx(t,s) ∂x(t,s) ∂x(t,s) ds ∂x(t,s) = + = = v (17a) dt ∂t ∂s dt ∂t x dy(t,s) ∂y(t,s) = = v (17b) dt ∂t y dz(t,s) ∂z(t,s) = = v (17c) dt ∂t z while noting that the variable s is independent of time (ds/dt =0) since it varies from 0 to 1 independently of t. The chain rule in (16) holds for any dx, dy, and dz so we can choose (x,y,z) to be the coordinates [x(t,s),y(t,s),z(t,s)] of the curve whose length has been normalized to 1. If we proceed as Maxwell did, using B = ∇× A to substitute ∂Ay/∂x = ∂Ax/∂y + Bz and ∂Az/∂x = ∂Ax/∂z − By into the third line of (15), we get for that line

1 ∂y ∂z ∂Ax ∂x ∂Ax ∂y ∂Ax ∂z ∂x Bz − By + + + ds I  ∂s ∂s ∂x ∂s ∂y ∂s ∂z ∂s  ∂t 0 1 ∂y ∂z ∂Ax ∂x = Bz − By + ds. (18) I  ∂s ∂s ∂s  ∂t 0 Because 2 ∂Ax ∂x ∂ x ∂ ∂x + Ax = Ax (19) ∂s ∂t ∂s∂t ∂s  ∂t  is a perfect differential, its integral around the closed curve of unity length is zero. Thus, (18), along with the similar expressions for the fourth and fifth lines in (15), reduce (15) to 1 d ∂Ax(r,t) ∂z ∂y ∂x A(r,t) · dl = + By − Bz dt I I  ∂t ∂t ∂t  ∂s C(t) 0

∂Ay(r,t) ∂x ∂z ∂y + + Bz − Bx  ∂t ∂t ∂t  ∂s ∂Az(r,t) ∂y ∂x ∂z + + Bx − By ds  ∂t ∂t ∂t  ∂s 

∂ = A(r,t) − v(r,t) × B(r,t) · dl (20) I ∂t  C(t) where v = ∂x/∂t ˆx + ∂y/∂t ˆy + ∂z/∂t ˆz. Consequently, Maxwell has proven that ∂ Ev(r,t) · dl = − A(r,t) − v(r,t) × B(r,t) · dl (21) I I ∂t  C(t) C(t) and, thus, he concludes that ∂ E (r,t)= − A(r,t) − ∇ψ (r,t)+ v(r,t) × B(r,t) (22) v ∂t e or in accordance with (10) (and our present-day notation E for the electric field)

Ev(r,t)= E(r,t)+ v(r,t) × B(r,t) (23) since Ev in (22) with v =0 has to equal E in (10). As mentioned above, Maxwell writes E = −∂A/∂t − ∇ψe explicitly in Article 783 for time varying fields with v = 0. It should be emphasized that Maxwell uses his mathematical formulation of Faraday’s law to obtain (22) and not the conductor current force density J × B in (1) that he has found from the force on a magnetic-shell model of circulating electric current. Therefore, Maxwell has been able to represent Faraday’s experimental results in a general mathematical form of Faraday’s law given in (6) with Ev given in (22) [1, arts. 598–599]. In one magnificent synthesis of mathematical and physical insight, he has not only put Faraday’s law on a solid mathematical foundation but he has also derived the “Lorentz force” for a moving unit electric charge. It is also possible to prove (6) and (22)–(23) from (8) using the Helmholtz transport theorem [12, ch. 6] of vector calculus, but Maxwell does not do this even though he mentions Helmholtz’s work with moving circuits in Article 544. Effectively, he proves the Helmholtz transport theorem for the electromagnetic fields in his mathematical formulation of Faraday’s law as part of his derivation reproduced above in (13)–(22). After deriving (22) from (6) in Article 598, he says that Ev(r,t) in (22) is the most general form of the electromotive force on a hypothetical unit point electric charge moving with C(t) at r, “being the force which would be experienced by a [moving] unit of positive electricity [electric charge] at that point.” It follows from linear superposition that the force on an electric charge q moving through electromagnetic fields is given by (in our present-day notation) F = q(E + v × B) (24) what we refer to today as the Lorentz force. Redˇzi´c[7] suggests that because “Maxwell’s ψe(r,t) does not have the same connotation as today’s scalar potential” that satisfies the Lorenz-Lorentz gauge, Maxwell may not have interpreted −∂A(r,t)/∂t − ∇ψe(r,t) as today’s electric field vector E(r,t) being the force that would be exerted on a hypothetical stationary unit electric charge placed at the point r. This is highly unlikely given that Maxwell defines the static electric field E(r) in Article 68 as the force that would be exerted on a hypothetical stationary unit electric charge at the point r and that Maxwell refers to Article 68 in his Article 598. Moreover, Maxwell always used the Coulomb gauge ∇ · A = 0 [2, secs. 6.2 and 6.4] and, thus, his scalar potential 2 always satisfies ∇ ψe(r,t) = −ρ(r,t)/ǫ [1, art. 783] but his electric force on a stationary unit electric charge is still given by E(r,t)= −∂A(r,t)/∂t − ∇ψe(r,t), a relationship that holds independently of the gauge and that Maxwell writes down explicitly for time varying fields in Article 783. Although Maxwell explained in Article 599 that the vector called Ev(r,t) herein is also the force experienced by the electric-polarization and conduction charges of a material body (which could be the ether) moving with the curve C(t) as confirmed by his writing in Articles 608, 609, and 619 that D = ǫEv (Maxwell looked at 2 2 D as electric polarization [13]) and J = σEv (correct for v /c ≪ 1), it seems clear from what Maxwell wrote in Articles 598 and 599 that he fully realized that −∂A(r,t)/∂t − ∇ψe(r,t) was the force exerted on a hypothetical stationary unit electric charge placed at the point r and that −∂A(r,t)/∂t − ∇ψe(r,t)+ v × B was the force exerted on a hypothetical moving (with velocity v) unit electric charge placed at the point r, even though he did not explicitly write Ev(r,t)= E(r,t)+v(r,t)×B(r,t) as is done in (23). For example, near the end of Article 598, Maxwell says that “Hence we may now disregard the circumstance that ds forms part of a circuit, and consider it simply as a portion of a moving body, acted on by the electromotive force [my Ev]. The electromotive force has already been defined in Art. 68. It is also called the resultant electrical force, being the force which would be experienced by a unit of positive electricity [electric charge] placed at that point. We have now obtained the most general value of this quantity in the case of a body moving in a magnetic field due to a variable electric system.” He continues in Article 599 with, “The electromotive force [on a particle with unit electric charge], the components of which are defined by equations (B) [(22) above], depends on three circumstances. The first of these is the motion of the particle [carrying unit electric charge] through the magnetic field [B]. The part of the force depending on this motion is expressed by the first two terms on the right of each equation [v × B in (22)].” From these and other statements in his Treatise, it seems clear that Maxwell realized that he had derived the force exerted by the electromagnetic fields in free space (ether) or conductors or polarized material7 at each instant of time on a hypothetical moving particle at r carrying unit electric charge8 and, if the velocity v of the circuit were zero, the measured time varying force at a point r on C would be the time varying electric field E(r,t)= −∂A(r,t)/∂t − ∇ψe(r,t). Moreover, he explicitly writes this equation for time varying fields and v = 0 in Article 783. Indeed, if this were not the case, it would mean that for stationary circuits C, Maxwell’s vector E(r,t) in his equation (6) of Faraday’s law would not refer to the time varying electric field as measured by the force on a hypothetical stationary unit electric charge placed at r. Maxwell used the same symbol “E” for the force on a unit electric charge whether or not the charge was moving. This does not conform to our present-day notation for the electric field but it does not represent a mistake in either his equations (reproduced herein) derived from Faraday’s law or his physical interpretation of these equations in terms of the force on moving and stationary unit electric charges. Other scientists used the same symbol for the force on a moving or stationary unit charge many years after Maxwell’s Treatise, notably, Poynting in his classic paper deriving what is now known as Poynting’s theorem [15], [16].

IV. CONCLUSION From the experiments of Faraday, Maxwell infers that the integral of the electromotive force around a moving circuit (closed curve) C(t) is given by the negative time derivative of the magnetic flux through any open surface S(t) bounded by the closed curve C(t). Using this generalized mathematical formulation of Faraday’s law given in (6), Maxwell effectively derives a Helmholtz transport theorem for the electromagnetic fields to prove that the electromagnetic force on a moving electric charge is given by the “Lorentz force” in (24) (using our present-day notation). This remarkable result derived in Maxwell’s Treatise can be traced back to his 1861 paper [8], which was written about 30 years before Heaviside [3] and Lorentz [4] expressed the force on a moving electric charge.

ACKNOWLEDGMENT The manuscript benefited from helpful discussions with Professor Dragan V. Redˇzi´c(who does not fully concur in [7] with some of the views presented here). This work was supported in part under the U.S. Air Force Office of Scientific Research Contract # FA9550-19-1-0097 through Dr. Arje Nachman.

REFERENCES [1] J.C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Edition, New York: Dover, 1954. The Dover edition is an unabridged, slightly altered, republication of the third edition, published by the Clarendon Press in 1891. All additions to the Treatise made by W. D. Niven and J. J. Thomson are ignored in [2] and in the present paper so as to concentrate on Maxwell’s original contributions. [2] A.D. Yaghjian, “Reflections on Maxwell’s Treatise,” PIER, 149, pp. 217–249, November 2014; see also “An overview of Maxwell’s Treatise,” FERMAT, 11, Sept.-Oct. 2015. [3] O. Heaviside, “On the electromagnetic effects due to the motion of electrification through a dielectric,” Phil. Mag. and J. Sci., fifth series, 27, pp. 324–339, 1889.

7Maxwell does not deal with the question of how a hypothetical moving electric charge placed instantaneously at r could measure the force E + v × B for C(t) in a polarized material. In magnetic polarization (magnetization), he determines his mathematically defined (macroscopic) magnetic fields [B, H] from his primary free-space magnetic field H0 measured in small cavities. He does not give an analogous prescription for determining the mathematically defined electric field from cavity fields in polarized dielectrics since Maxwell considered D to be the electric polarization and did not introduce a polarization vector P [2], [13]. 8Even though Maxwell derived the force on electric charge moving through electromagnetic fields, apparently most of the scientific community did not understand what he had done until much later when Lorentz used this force extensively in his work. In fact, Maxwell’s equations were not widely accepted until after Hertz experimentally demonstrated the existence of wireless microwave radiation [14]. This reluctance to accept Maxwell’s equations until well after he had died, Maxwell’s direct but mathematically sophisticated deduction of the v × B force in Faraday’s law in Article 598, and his use of potentials and the same symbol E for the force on both a stationary and moving unit electric charge are probably the main reasons that Maxwell’s derivation of the “Lorentz force” continues to be largely overlooked. [4] H.A. Lorentz, “La th´eorie ´electromagn´etique de Maxwell et son application aux corps mouvants,” Archives Neerlandaises´ des Sciences Exactes et Naturelles, 25, pp. 363–552, 1892. [5] J.D. Buchwald, From Maxwell to Microphysics, Chicago: University of Chicago Press, 1985. [6] M. Faraday, Experimental Researches in Electricity, New York: Dover, 2004; Originally published in three volumes by J.E. Taylor: London, 1839–1855. [7] D.V. Redˇzi´c, “Maxwell’s inductions from Faraday’s induction law,” Eur. J. Phys., 39, 025205 (16pp), February 2018. [8] J.C. Maxwell, “On physical lines of force, Part 2” Phil. Mag. and J. Sci., fourth series, 21, pp. 282–349, March 1861. [9] J.C. Maxwell, “A dynamical theory of the electromagnetic field” Phil. Trans. Roy. Soc. Lond., 155, pp. 459–512, 1865. [10] O.M. Bucci, “The genesis of Maxwell’s equations,” ch. 4 in History of Wireless, Hoboken, NJ: Wiley, 2006. [11] J.C. Maxwell, “On Faraday’s lines of force,” Trans. Cambridge Phil. Soc., 10, pp. 27–83, 1856. [12] C.-T. Tai, Generalized Vector and Dyadic Analysis, New York: IEEE/Wiley, 1997. [13] A.D. Yaghjian, “Maxwell’s definition of electric polarization as displacement,” PIER-M, 88, pp. 67-71, January 2020. [14] H. Hertz, “Uber¨ sehr schnelle electrische Schwingungen,” plus two other papers, Annalen der Physik, 31, new series, pp. 421–448; 543–544; 983–1000, 1887. [15] J.H. Poynting,“On the transfer of energy in the electromagnetic field,” Phil. Trans. Roy. Soc. Lond., 175, pp. 343–361, January 1884. [16] A.D. Yaghjian, “Classical power and energy relations for macroscopic dipolar continua derived from the microscopic Maxwell equations,” PIER B, 71, pp. 1–37, 2016.